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Subdivided Graphs As Isometric Subgraphs of Hamming Graphs Laurent Beaudou, Sylvain Gravier, Kahina Meslem

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Subdivided Graphs As Isometric Subgraphs of Hamming Graphs Laurent Beaudou, Sylvain Gravier, Kahina Meslem Subdivided graphs as isometric subgraphs of Hamming graphs Laurent Beaudou, Sylvain Gravier, Kahina Meslem To cite this version: Laurent Beaudou, Sylvain Gravier, Kahina Meslem. Subdivided graphs as isometric subgraphs of Hamming graphs. European Journal of Combinatorics, Elsevier, 2009, 30, pp.1062–1070. 10.1016/j.ejc.2008.09.011. hal-00192300 HAL Id: hal-00192300 https://hal.archives-ouvertes.fr/hal-00192300 Submitted on 27 Nov 2007 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Institut Fourier Institut Fourier Unit´eMixte de Recherche 5582 CNRS – Universit´eJoseph Fourier Subdivided graphs as isometric subgraphs of Hamming graphs Laurent Beaudou, Sylvain Gravier and Kahina Meslem 10 October 2007 Subdivided graphs as isometric subgraphs of Hamming graphs Laurent Beaudou, Sylvain Gravier and Kahina Meslem Institut Fourier UMR 5582 CNRS – Universit´eJoseph Fourier Institut Fourier 100, rue des Maths 38402 St-Martin d’H`eres – FRANCE [email protected] [email protected] kahina [email protected] Abstract/R´esum´e It is proven that given G a subdivision of a clique Kn (n ≥ 1), G is isometrically embeddable in a Hamming graph if and only if G is a partial cube or G = Kn. The characterization for subdivided wheels is also obtained. Keywords: isometric embeddings, subdivided graph, partial cube, Hamming graph. Dans ce rapport, on prouve qu’´etant donn´e G une clique Kn (n ≥ 1) subdivis´ee, G est le sous-graphe isom´etrique d’un graphe de Hamming si et seulement si G est un cube partiel ou G = Kn. La caract´erisation est aussi obtenue pour les roues subdivis´ees. Mots-cl´es : plongement isom´etrique, graphe subdivis´e,cube partiel, graphe de Hamming. Introduction Isometric subgraphs of Hamming graphs (resp. hypercubes) are called partial Hamming graphs (resp. partial cubes). Partial cubes have first been investigated by Graham and Pollak [11], and Djokovi`c[9]. Later, several algorithmic charac- terizations were shown using a relation defined on the set of edges (the Θ relation introduced by Djokovi´c[9] and Winkler [18]), or constructive operations. This relation easily led to a recognition algorithm for partial cubes running in O(mn) (m is the number of edges and n the number of vertices). This complexity was then improved for specific classes of graphs by Breˇsar et al. [3]. And recently, the general complexity was pulled down to O(n2) by Eppstein [10]. Partial cubes have found different applications, for instance, in [6, 8, 13], interesting applications in chemical graph theory were established. The search for structural characterizations was first motivated by a conjec- ture of Chepoi and Tardif in the 1994 Bielefeld conference on “Discrete Metric Spaces”. They asked if bipartite graphs with convex intervals were partial cubes. The negative answer was brought by Breˇsar and Klavˇzar in [4], using subdivided wheels. From this point structural characterizations for partial cubes that were subdivisions of wheels and cliques were obtained (see [1, 2, 12]). A natural extension of this work was to study partial Hamming graphs. One point of interest about partial Hamming graphs, is that they can be labelled in such a way that, given a target vertex, only local information is needed to know which way is a shortest path. Reader can see [13, 14, 15, 17, 18] for further information about partial Hamming graphs. Moreover, studying subdivided graphs and their isometric embeddings in Hamming graphs, allows us to study partly the λ-scale embeddings in Hamming graphs. For example, if the subdivision of a graph G is isometrically embeddable in a hypercube, then G is 2-scale embeddable in a hypercube (see [7, 16] for more details). In this paper, we show that, the only subdivision of a clique isometrically embeddable in a Hamming graph which is not a partial cube ; is actually the clique itself. Then, we study the subdivisions of wheels that are isometrically embeddable in Hamming graphs and obtain a structural characterization of them. 1 Preliminary definitions and basic properties By replacing edges of a graph G by paths, we obtain a subdivided graph of G. Given G a graph, S(G) is obtained by subdividing each edge of G exactly once. We call principal vertex, a vertex of a subdivided graph of G that was not added during the subdivision (it was already in G). We call universal vertex, a vertex of a subdivided graph of G such that each edge incident to it, is not subdivided. The Cartesian product G¤H of graphs G and H is the graph with vertex set V (G)×V (H) in which the vertex (a, x) is adjacent to the vertex (b, y) whenever ab ∈ E(G) and x = y, or a = b and xy ∈ E(H). A Hamming graph is a cartesian product of cliques. For a graph G, the distance dG(u,v) between vertices u and v is defined as the number of edges on a shortest uv-path. A subgraph H of G is called 3 isometric if and only if dG(u,v) = dH (u,v) for all u,v ∈ V (H). For k ≥ 3, the k-fan Fk is the graph obtained as the join of a vertex u and a path on k vertices w1,...,wk (see Fig. 1(a)). For k ≥ 3, the k-wheel Wk is the graph obtained as the join of a vertex u and a cycle on k vertices w1,...,wk (see Fig. 1(b)). w3 u u w4 w2 w1 w4 w2 w3 w5 w1 (a) Subdivision of W5 (b) Fan F4 Figure 1: We also remind some former results about partial cubes. Lemma 1. [12] Let G be a graph and K an isometric subgraph of G which is isomorphic to a subdivision of Fk (k ≥ 3) such that at least one of the edges uw2,...,uwk−1 is subdivided. Then G is not a partial cube. Theorem 2. [12] A subdivided wheel (with more than 4 rays) is a partial cube if and only if all its rays are not subdivided and the other edges are oddly sub- divided. Theorem 3. [2] Let G be a subdivided graph of a complete graph Kn (n ≥ 4). G is a partial cube if and only if, G is isomorphic to S(Kn) or G contains a universal vertex u and the number of added vertices to each edge not incident to u in Kn is odd. Given G a partial Hamming graph and an isometric embedding φ of G into a Hamming graph H, φ induces a natural edge labelling ℓ of G : given e an edge of G linking two vertices u and v, as they are adjacent in G and φ is isometric, we get that dH (φ(u), φ(v)) is 1 and thus φ(u) and φ(v) differ in exactly one coordinate i. We define ℓ(e) = i. xi will denote the i-th coordinate of the vertex x embedded in a Hamming graph. Lemma 4. Given G a partial Hamming graph and a natural labelling of its edges ℓ. If two edges of G, e and f are part of a geodesic path in G, then ℓ(e) 6= ℓ(f). Proof. A shortest path in a Hamming graph cannot use twice the same coordi- nate, or it would be shorter using it once or not using it at all. Lemma 5. Given G a partial Hamming graph and a natural labelling of its edges ℓ. Given C an elementary cycle of G, each label appearing in C, appears at least twice. Moreover, if C is isometric and differs from K3, then these labels appear exactly twice. 4 Proof. For a contradiction let us suppose there is a cycle (x0,x1,...,xk,x0) such i i that ℓ(x0x1) = i and no other edge of the cycle has label i. Then x0 6= x1 (since i i ℓ(x0x1) = i). For all other edges, xl = xl+1 (1 ≤ l ≤ k − 1) which implies that i i i i xk = x1. The last edge (xkx0) leads to a contradiction since x0 6= x1. Remark 6. It is straightforward from this Lemma that an isometric cycle in a partial Hamming graph is either even or K3. x z t y Figure 2: K4 − e Proposition 7. K4−e (Fig. 2) cannot be isometrically embedded in a Hamming graph. Proof. x and y differ on exactly one coordinate i, we can assume that xi = 0 and yi = 1. Consequently, all the vertices that are at distance 1 from x and y, differ also on the same coordinate i. Therefore, they are at distance 1 from each other. Thus, t and z should be linked, which is not the case. 2 Structural characterization for cliques We shall prove the following theorem. Theorem 8. Given G a subdivided clique Kn for some n, G is a partial Ham- ming graph if and only if G is a partial cube or G = Kn 2.1 First case : G is bipartite Let us show that only two values are used for each coordinate. For a contradiction, suppose that there exists a coordinate i and three ver- tices x,y,z such that xi = 0, yi = 1 and zi = 2.
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